I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are closed under addition, multiplication and exponentiation.
What are the quaternions and octonions (I remember seeing $j$ and $k$) and other hypercomplex systems (as they are called), and why did we create them?
Also, I remember reading that the octonions are the largest of these hypercomplex systems (meaning that any number in a hypercomplex system is also a number in the system of the octonions).
Thank you very much in advance.
The place to look is John Baez's beautifully written article on The Octonions. The introduction is wonderfully entertaining and the relevant section you want to focus on is the Cayley-Dickson construction.